Can I use iCloud away from home on my iPhone5s?

I thought that it worked that way, but I seem to be mistaken. I can use it when on the same block as my home, but not otherwise. When I am near home, it only allows me to play a few songs from a couple of artists. Is it supposed to work that way and if not, how do I correct it? I don't know if it affects anything, but I have Verizon and the standard 5GB cloud plan (but am willing to upgrade if it will help my listening coverage). I've checked several similar questions but have yet to find anything that works. This is a big annoyance for me because I wanted to listen to my songs away from home, but my 16GB phone won't hold all 1400 songs (which I'm told is another problem all it's own, but that's another question entirely).